Asimple first-price auction: an item is for sale in an auction. player 1 values the item at 3 while player 2 values the item at 5. each player can bid 0, 1, or 2. if player i bids more that player j, then player i wins the item and pays his bid while player j loses and pays 0 and has a zero payoff. the winners’ payoff is his or her value less his or her bid. if the players bid the same amount, then a coin toss determines the winner, and winner gets the item and the loser does not. this is true even if both bid 0. when the bids are the same calculated the payoffs as the average of 0, the payoff of the loser, and the player’s valuation less the bid, the payoff of the winner. more precisely, when there is a ties the chance of each player winning is 50%, and you should calculate the expected payoff or the probability-weighted payoff.

Pick the expression that matches this description: a polynomial of the 5th degree with a leading coefficient of 7 and a constant term of 6 (choice a) 7x^5+2x^2+6 (choice b) 7x^6−6x^4+5 (choice c) 6x^7−x^5+5 (choice d) 6x^5+x^4+7

Scores on a university exam are normally distributed with a mean of 78 and a standard deviation of 8. the professor teaching the class declares that a score of 70 or higher is required for a grade of at least “c.” using the 68-95-99.7 rule, what percentage of students failed to earn a grade of at least “c”?

Each of the following data sets has a mean of x = 10. (i) 8 9 10 11 12 (ii) 7 9 10 11 13 (iii) 7 8 10 12 13 (a) without doing any computations, order the data sets according to increasing value of standard deviations. (i), (iii), (ii) (ii), (i), (iii) (iii), (i), (ii) (iii), (ii), (i) (i), (ii), (iii) (ii), (iii), (i) (b) why do you expect the difference in standard deviations between data sets (i) and (ii) to be greater than the difference in standard deviations between data sets (ii) and (iii)? hint: consider how much the data in the respective sets differ from the mean. the data change between data sets (i) and (ii) increased the squared difference îł(x - x)2 by more than data sets (ii) and (iii). the data change between data sets (ii) and (iii) increased the squared difference îł(x - x)2 by more than data sets (i) and (ii). the data change between data sets (i) and (ii) decreased the squared difference îł(x - x)2 by more than data sets (ii) and (iii). none of the above